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   "source": [
    "## Least-squares solutions in Julia.\n",
    "\n",
    "<br />\n",
    "\n",
    "When we're confronted with **large amounts of data**, we often look for a **simple quantitative model** that describes basic trends in the data.\n",
    "\n",
    "<br />\n",
    "\n",
    "For example, suppose we are measuring the amount of time customers spend in our store. We want to understand the relationship between variables $t=$\"time spent in store\" and $m$=\"money spent at checkout.\"\n",
    "\n",
    "<br />\n",
    "\n",
    "Here is a sample of (synthetic) data collected in a matrix: the left column has the time measurements $t_1,t_2,\\ldots,t_n$ and the right column has the amount of measurements $m_1,m_2,\\ldots,m_n$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "id": "f8374dc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100×2 Matrix{Float64}:\n",
       " 13.2019    112.634\n",
       "  7.81818    61.812\n",
       "  8.07477    47.5651\n",
       " 19.1703    147.968\n",
       "  1.50192    21.4934\n",
       " 18.2696    152.165\n",
       "  9.45868    56.4961\n",
       "  3.2809     45.5131\n",
       "  7.22552    59.3135\n",
       "  8.81721    62.3885\n",
       "  2.99229    21.4012\n",
       "  2.63958    35.0781\n",
       " 15.5887    101.839\n",
       "  ⋮         \n",
       "  0.668849   32.6133\n",
       " 11.8905     98.7587\n",
       " 11.1965     68.2214\n",
       " 15.7339    134.037\n",
       "  1.13775    45.1733\n",
       " 15.5209    128.961\n",
       " 19.6157    165.514\n",
       " 10.1839     84.5181\n",
       " 18.7281    136.218\n",
       " 15.4049    104.055\n",
       " 16.6003    138.31\n",
       " 16.1531    112.939"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# generate customer data\n",
    "n = 100\n",
    "t = 20*rand(n)\n",
    "#m = 0.3*t.^2 + 2*t + 30*rand(n)\n",
    "m = 300 ./ (1 .+ exp.(-(t.-20)/6.5)) + 35*rand(n)\n",
    "\n",
    "data = [t m]"
   ]
  },
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   "source": [
    "We can visualize the relationship between the columns better by plotting the data in the $(t,m)$ plane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "id": "31b3653c",
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     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "# plot the data\n",
    "plot(t, m, seriestype=:scatter, label=:\"data\")\n",
    "xlabel!(\"time\")\n",
    "ylabel!(\"money\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "536610d2",
   "metadata": {},
   "source": [
    "Plotting by eye is a great tool, but can we write down a simple equation that describes the relationship between $t$ and $m$? An equation is useful if we want to make predictions about the behavior of future customers.\n",
    "\n",
    "<br />\n",
    "\n",
    "## Linear regression\n",
    "\n",
    "Let's start by finding the line that best fits the data. We need to find the slope $D$ and intercept $C$ that make the two sides of these equations agree:\n",
    "\n",
    "$$ \\begin{aligned} C + Dt_1 &= m_1, \\\\ C + Dt_2 &= m_2, \\\\ \\vdots \\\\ C + Dt_n &= m_n. \\end{aligned} $$\n",
    "\n",
    "We have $n$ equations in $2$ unknowns, $C$ and $D$. In matrix notation, the equations become $Ax=m$:\n",
    "\n",
    "$$ \\begin{pmatrix}1 & t_1 \\\\ 1 & t_2 \\\\ \\vdots & \\vdots \\\\ 1 & t_n  \\end{pmatrix}\\begin{pmatrix} C \\\\ D\\end{pmatrix} = \\begin{pmatrix}m_1 \\\\ m_2 \\\\ \\vdots \\\\ m_n \\end{pmatrix}. $$\n",
    "\n",
    "If $t_1,t_2,\\ldots,t_n$ are not all the same, then $A$ has independent columns. The equations usually have no solution because we can't make a single straight line go through all the data points. The right-hand side is usually _not_ in the column space. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "id": "98109c1a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "A = [ones(n) t]             # coefficient matrix\n",
    "display(rank(A))            # A usually has rank(A) = 2 independent columns\n",
    "display(rank([A m]))        # b is usually not a combination of the columns of A"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "2659fe02",
   "metadata": {},
   "source": [
    "Instead, we choose $x$ to make $Ax-b$ as small as possible! This special value $x=x_*$ is called the _least-squares solution_. How do we find $x_* = [C_{\\rm best},\\,\\, D_{\\rm best}]^T$? We solve the normal equations!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "id": "13ebd345",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " 11.954859857331227\n",
       "  6.928107852700091"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = A' * A\n",
    "b = A' * m\n",
    "xstar = M \\ b"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "c1d5cd8f",
   "metadata": {},
   "source": [
    "The best intercept is $C_{\\rm best}\\approx -17.5$ and the best slope is $D_{\\rm best}\\approx 10.2$. Let's plot the line over our data to see how it looks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "f536e23a",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tt = 20*collect(0:250)/250      # equally spaced points in the interval [0, 20]\n",
    "mm = xstar[1] .+ xstar[2]*tt\n",
    "\n",
    "plot!(tt,mm, lc=:black, lw=:3)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "90a0ab3b",
   "metadata": {},
   "source": [
    "Not bad! Our best fit line is clearly capturing the basic trend in the data. But to my eye, comparing to a straight line reveals some curvature in the data. Can we capture this curvature quantitatively?\n",
    "\n",
    "<br />\n",
    "\n",
    "## Quadratic regression\n",
    "\n",
    "We try to caputre the upward curve by adding a quadratic term to our linear model. We need to find the polynomial coefficients in $p(x)=D+Ct+Et^2$ that make the two sides of these equations agree:\n",
    "\n",
    "$$ \\begin{aligned} C + Dt_1 + Et_1^2 &= m_1, \\\\ C + Dt_2 + Et_2^2 &= m_2, \\\\ \\vdots \\\\ C + Dt_n + Et_n^2 &= m_n. \\end{aligned} $$\n",
    "\n",
    "We now have $n$ equations in $3$ unknowns, $C$, $D$, and $E$, and in matrix notation,\n",
    "\n",
    "$$ \\begin{pmatrix}1 & t_1 & t_1^2\\\\ 1 & t_2 & t_2^2\\\\ \\vdots & \\vdots \\\\ 1 & t_n & t_n^2  \\end{pmatrix}\\begin{pmatrix} C \\\\ D \\\\ E\\end{pmatrix} = \\begin{pmatrix}m_1 \\\\ m_2 \\\\ \\vdots \\\\ m_n \\end{pmatrix}. $$\n",
    "\n",
    "Again, the equations usually have no solution because we can't make a single straight line go through all the data points, i.e., the right-hand side is usually _not_ in the column space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "id": "3da26032",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [ones(n) t t.^2]        # new n x 3 coefficient matrix\n",
    "display(rank(A))            # A usually has rank(A) = 3 independent columns\n",
    "display(rank([A m]))        # b is usually not a combination of the columns of A"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "57252ae6",
   "metadata": {},
   "source": [
    "Instead, we solve the normal equations and find the least-squares solution $x_*=[C,\\,\\,D,\\,\\,E]^T$, the coefficients of the quadratic polynomial that best fits the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "id": "d2385daa",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve the normal equations\n",
    "M = A' * A\n",
    "b = A' * m\n",
    "xstar = M \\ b\n",
    "\n",
    "# visualize the least-squares fit at equispaced points in [0 20]\n",
    "tt = 20*collect(0:250)/250\n",
    "mm = xstar[1] .+ xstar[2]*tt .+ xstar[3]*tt.^2\n",
    "\n",
    "# plot the data\n",
    "plot(t, m, seriestype=:scatter, label=:\"data\")\n",
    "plot!(tt, mm, lc=:black, lw=:3)\n",
    "xlabel!(\"time\")\n",
    "ylabel!(\"money\")\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "28279210",
   "metadata": {},
   "source": [
    "Even better! It looks like we captured the \"centerline\" of the data pretty well - the data points are roughly equally distributed on both sides of the line.\n",
    "\n",
    "## Model predictions\n",
    "\n",
    "So, can we use our least-squares fit to make predictions? What if we want to make a prediction about customers who shop longer: $20$ to $40$ minutes?\n",
    "\n",
    "<br /> \n",
    "\n",
    "Let's get some more customer data, and compare with our quadratic model prediction..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "id": "b581716e",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# generate customer data\n",
    "n = 200\n",
    "t = 40*rand(n)\n",
    "#m = 0.3*t.^2 + 2*t + 30*rand(n)\n",
    "m = 300 ./ (1 .+ exp.(-(t.-20)/6.5)) + 35*rand(n)\n",
    "\n",
    "\n",
    "# visualize the least-squares fit preduction at equispaced points in [0 40]\n",
    "tt = 40*collect(0:500)/500\n",
    "mm = xstar[1] .+ xstar[2]*tt .+ xstar[3]*tt.^2\n",
    "\n",
    "# plot the data\n",
    "plot(t, m, seriestype=:scatter, label=:\"data\")\n",
    "plot!(tt, mm, lc=:black, lw=:3)\n",
    "xlabel!(\"time\")\n",
    "ylabel!(\"money\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "e924935e",
   "metadata": {},
   "source": [
    "Our model appears to get worse as we move from $20$ minutes to $40$ minutes, and seriously overestimates how much the long shoppers are spending.\n",
    "\n",
    "<br />\n",
    "\n",
    "Be warned: using regression to _extrapolate_ from data can produce disastrous predictions away from the region where the data was collected! \n",
    "\n",
    "<br />\n",
    "\n",
    "## What should we do?\n",
    "\n",
    "<br />\n",
    "\n",
    "Your ideas go here!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "29841328",
   "metadata": {},
   "outputs": [],
   "source": [
    "## Build a better model\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## Build a better model"
   ]
  }
 ],
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